Problem: Simplify the following expression: $ n = \dfrac{5z + 6}{-3z - 1} - \dfrac{1}{9} $
Solution: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{9}{9}$ $ \dfrac{5z + 6}{-3z - 1} \times \dfrac{9}{9} = \dfrac{45z + 54}{-27z - 9} $ Multiply the second expression by $\dfrac{-3z - 1}{-3z - 1}$ $ \dfrac{1}{9} \times \dfrac{-3z - 1}{-3z - 1} = \dfrac{-3z - 1}{-27z - 9} $ Therefore $ n = \dfrac{45z + 54}{-27z - 9} - \dfrac{-3z - 1}{-27z - 9} $ Now the expressions have the same denominator we can simply subtract the numerators: $n = \dfrac{45z + 54 - (-3z - 1) }{-27z - 9} $ Distribute the negative sign: $n = \dfrac{45z + 54 + 3z + 1}{-27z - 9}$ $n = \dfrac{48z + 55}{-27z - 9}$ Simplify the expression by dividing the numerator and denominator by -1: $n = \dfrac{-48z - 55}{27z + 9}$